Polynomials on n-sphere as a reproducing kernel Hilbert Space

Question

In "On the Mathematical foundations of Learning", p. 38, Cucker takes the example of the vector space $H_d$ of homogeneous polynomials of degree $d$ on sphere $\mathbb{S}^n = S(\mathbb{R}^{n+1})$ with inner product: $$ \left\langle \sum_{\alpha \in A_d} w_\alpha \cdot x^\alpha, \sum_{\alpha \in A_d} w'_\alpha \cdot x^\alpha \right\rangle_{H_d} = \sum_{\alpha \in A_d} w_\alpha \cdot w'_\alpha \cdot \frac{\alpha!}{d!} $$ where $A_d = \{ \alpha \in \mathbb{N}^{n+1} : \alpha_0 + \cdots + \alpha_n = d \}$, $x^\alpha = x_0^{\alpha_0} \cdots x_n^{\alpha_n}$, and $\alpha! = \alpha_0 ! \cdots \alpha_n !$.

He says one can prove $H_d$ is also the Hilbert Space associated to the reproducing kernel $K(x,x') = \left( x_0 \cdot x'_0 + \cdots + x_n \cdot x'_n \right)^d$ with $x$ and $x'$ in $\mathbb{S}^n$.

Now Mercer's theorem says that $K$ can be written $K(x,x') = \sum_{\alpha \in A_d} \lambda_\alpha \cdot \phi_\alpha(x) \cdot \phi_\alpha(x')$ with a family of positive eigenvalues $(\lambda_\alpha)_{\alpha \in A_d}$ and a family $(\phi_\alpha)_{\alpha \in A_d}$ of eigenvectors in $L^2(\mathbb{S}^n)$ orthonormal for the usual inner product: $$\langle f, f' \rangle_{L^2(\mathbb{S}^n)} = \int_{x \in \mathbb{S}^n} f(x) \cdot f'(x) \cdot dx$$

Given that the feature map $\Phi(x) = (\sqrt{\lambda_\alpha} \cdot \phi_\alpha(x))_\alpha$ is according to Cucker $\Phi(x) = (\sqrt{ d! / \alpha !} \cdot x^\alpha)_\alpha$, it seems the eigenvectors should be something like $\phi_\alpha(x) \propto x^\alpha$. But I cannot understand how such a family could be orthonormal for $ \langle \;, \; \rangle_{L^2(\mathbb{S}^n)} $. For example in the case $n = 1$ and $d =4$: $$\langle \phi_{(1, 3)}, \phi_{(3, 1)} \rangle_{L^2(\mathbb{S}^1)} \propto \int_{x \in \mathbb{S}^1} x_1^4 \cdot x_2^4 \cdot dx > 0 $$

Am I considering the wrong inner product ?

Answer 1

@zack: It seems you were correct about your polynomials. I don't know for the Laplacian operator but at least they are also eigenvectors of the operator $f \mapsto \int \langle x, t \rangle^4 \cdot f(t) \cdot dt$.

In the end and unexpectedly enough the feature map Cucker provides is not of the form $ (\sqrt{\lambda_\alpha} \cdot \phi_\alpha(x))_\alpha$. I computed that for example with $K(x, x') = \langle x, x' \rangle^2$ on $\mathbb{S}^1$ we have:

• $\lambda_0 = \pi$ and $\phi_0 (x) = \frac{1}{\sqrt{2\pi}} \cdot (x_0^2+x_1^2)$
• $\lambda_1 = \pi/2$ and $\phi_1(x) = \frac{2}{\sqrt{\pi}} \cdot x_0 \cdot x_1$
• $\lambda_2 = \pi/2$ and $\phi_2(x) = \frac{1}{\sqrt{\pi}} \cdot (x_0^2-x_1^2)$

So the feature map should be: $$\Phi_{\textrm{expected}}(x) = \left( \frac{1}{\sqrt{2}}(x_0^2+x_1^2), \sqrt{2} \cdot x_0 \cdot x_1, \frac{1}{\sqrt{2}}(x_0^2-x_1^2) \right)$$ whereas the one provided is: $$\Phi_{\textrm{Cucker}}(x) = \left( x_0^2, \sqrt{2} \cdot x_0 \cdot x_1, x_1^2 \right)$$

The fact is we have $ \langle \Phi_{\textrm{expected}}(x), \Phi_{\textrm{expected}}(t) \rangle = \langle \Phi_{\textrm{Cucker}}(x), \Phi_{\textrm{Cucker}}(t) \rangle = K(x, t) $ so it seems several equivalent feature maps may be used. However this fact is a bit confusing and nowhere mentioned in the article.